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Chapter 5: Problem 18

Multiply. $$ \left(-\frac{1}{4} x^{4}\right)\left(\frac{1}{5} x^{8}\right) $$

Short Answer

Expert verified
-\frac{1}{20} x^{12}

Step by step solution

01

Multiply the Coefficients

First, multiply the coefficients of the given expressions. The coefficients are \(-\frac{1}{4}\) and \(\frac{1}{5}\). Therefore, \[ -\frac{1}{4} \times \frac{1}{5} = -\frac{1}{20} \]
02

Multiply the Variables with Exponents

Next, we need to multiply the variables \(x^4\) and \(x^8\). When multiplying variables with exponents, we add the exponents. Therefore, \[ x^4 \times x^8 = x^{4+8} = x^{12} \]
03

Combine Both Results

Combine the results from Step 1 and Step 2. The combined result will be the product of the coefficients and the product of the variables: Therefore, \[ -\frac{1}{20} \times x^{12} = -\frac{1}{20} x^{12} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Multiplying Coefficients
When multiplying polynomial expressions, the first step involves multiplying the coefficients. Coefficients are the numeric parts of the terms. For example, in the expression \(-\frac{1}{4} x^{4}\), the coefficient is \(-\frac{1}{4}\). In our exercise, we're multiplying \(-\frac{1}{4}\) and \(\frac{1}{5}\). This step is straightforward:
  • Multiply the coefficients: \(-\frac{1}{4} \times \frac{1}{5} = -\frac{1}{20}\)

This gives us the coefficient of the product. Ensure you multiply fractions correctly by multiplying the numerators and the denominators separately.
Adding Exponents
In polynomial multiplication, after dealing with the coefficients, we next focus on the variables with exponents. The rule here is to add the exponents when multiplying like bases.
  • If you have \(x^a\) and \(x^b\), you'll end up with \(x^{a+b}\).

For the given exercise, we multiply \(x^4\) by \(x^8\), resulting in:
  • Adding the exponents: \(x^{4 + 8} = x^{12}\)

Always ensure that the bases (the variables) are the same before adding the exponents.
Polynomial Multiplication Steps
Combining everything, let's go through the step-by-step process of multiplying polynomial expressions.
  • Step 1: Multiply the coefficients.
    We found: \(-\frac{1}{4} \times \frac{1}{5} = -\frac{1}{20}\)
  • Step 2: Multiply the variables with exponents.
    We found: \(x^4 \times x^8 = x^{12}\)
  • Step 3: Combine the results of Step 1 and Step 2 to get the product.
    This gives us: \(-\frac{1}{20} x^{12}\)

Thus, the final result of the multiplication is \(-\frac{1}{20} x^{12}\). Ensure to follow these steps for any polynomial multiplication problem. Practice with different examples to become more comfortable with each step.

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